Big old equestrian statue of the 1st Duke of Wellington, by Carlo Marochetti. Erected in 1844, and sits right in the front of the square, in front of the Glasgow Gallery of Modern Art, a building that used to be the city mansion of the tobacco baron William Cunninghame. Not many people know or care about any of this, and most people don't know who the bloke on the horse is supposed to be. It's only notable because years back, some joker climbed up and stuck a traffic cone on top so the duke's wearing it like a hat. This summarises Glasgow's sense of humour pretty nicely, and for as long as anyone can remember, the council have kept taking it down and people have kept putting it back up. Officially, Glasgow Council and Strathclyde Police are obliged to say that it's vandalism, since the statue's a listed monument and probably a little bit fragile by this point. As my esteemed colleague MrF points out, by this point, the duke's lost both spurs and half of his sword to drunken Weegies climbing on him. Still happens, though.
I know two stories about it. First one: I used to know a guy who swore blind that there was a whole network of tunnels under the square, and that the traffic cone was a secret Illuminati signal marking the existence of a giant secret bunker underneath it. Thus, every time you saw a traffic cone somewhere it wasn't supposed to be, it indicated conspiratorial activity. I've been meaning to write that up as fiction for months, and haven't found the right words yet.
Story two: One morning, when I was having a coffee and a smoke at Costa in the square, looking at the statue, I saw a council works van with a crane on it pull up, blocking half the traffic down Queen Street. A pair of council workers in hi-vis jackets got up in the crane next to the (at that point unconed) duke, plonked a fresh traffic cone atop his head, took a promotional photo of it for the council to demonstrate what a fun and happening place Glasgow was, then yanked it back off, got in the van and left, leaving a rather amused-looking crowd. Which is a pretty neat summary of Glasgow council's sense of humour.